Vector optimization

Vector optimization is a subarea of mathematical optimization where optimization problems with a vector-valued objective functions are optimized with respect to a given partial ordering and subject to certain constraints. A multi-objective optimization problem is a special case of a vector optimization problem: The objective space is the finite dimensional Euclidean space partially ordered by the component-wise "less than or equal to" ordering.

Problem formulation

In mathematical terms, a vector optimization problem can be written as:

C\operatorname{-}\min_{x \in S} f(x)

where f: X \to Z for a partially ordered vector space Z. The partial ordering is induced by a cone C \subseteq Z. X is an arbitrary set and S \subseteq X is called the feasible set.

Solution concepts

There are different minimality notions, among them:

Every proper minimizer is a minimizer. And every minimizer is a weak minimizer.[1]

Modern solution concepts not only consists of minimality notions but also take into account infimum attainment.[2]

Solution methods

Relation to multi-objective optimization

Any multi-objective optimization problem can be written as

\mathbb{R}^d_+\operatorname{-}\min_{x \in M} f(x)

where f: X \to \mathbb{R}^d and \mathbb{R}^d_+ is the non-negative orthant of \mathbb{R}^d. Thus the minimizer of this vector optimization problem are the Pareto efficient points.

References

  1. Ginchev, I.; Guerraggio, A.; Rocca, M. (2006). "From Scalar to Vector Optimization". Applications of Mathematics 51: 5. doi:10.1007/s10492-006-0002-1.
  2. 1 2 Andreas Löhne (2011). Vector Optimization with Infimum and Supremum. Springer. ISBN 9783642183508.
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